It is a problem when someone who is generally a pain suggests something that is not completely unreasonable!
Depending on what is in your By-Laws now, perhaps the way for the EC to deal with this is to propose a By-Law that would be a reasonable compromise on pets. If it gets the necessary support you will have more teeth to deal with complaints that contravene the By-Law. On the other hand if the majority of owners don’t support the proposal then you can shrug your shoulders when you get complaints and say you tried but can only act to reflect the resolutions of general meetings and on this one there was not the support.
By reasonable By-laws, I’d suggest such things as:
-members much seek approval for pets but approval will not be unreasonably refused (that is something like the ACT Unit Titles Act wording).
-Dogs must be on leash if on common property (perhaps you could designate an off leash area if that would be practical in your setting)
-Faeces must be removed by pet owners.
-Cats must not be allowed to roam but outdoor cat runs may be approved within unit areas.
Various lawyers have off the shelf pet by-laws you can purchase.